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 Algebra Logika, 2010, Volume 49, Number 6, Pages 819–833 (Mi al469)

Nilpotent length of a finite group admitting a Frobenius group of automorphisms with fixed-point-free kernel

E. I. Khukhro

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

Abstract: Suppose that a finite group $G$ admits a Frobenius group $FH$ of automorphisms with kernel $F$ and complement $H$ such that the fixed-point subgroup of $F$ is trivial, i.e., $C_G(F)=1$, and the orders of $G$ and $H$ are coprime. It is proved that the nilpotent length of $G$ is equal to the nilpotent length of $C_G(H)$ and the Fitting series of the fixed-point subgroup $C_G(H)$ coincides with a series obtained by taking intersections of $C_G(H)$ with the Fitting series of $G$.

Keywords: Frobenius group, automorphism, finite group, soluble group, nilpotent length, Fitting series.

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English version:
Algebra and Logic, 2010, 49:6, 551–560

Bibliographic databases:

UDC: 512.542

Citation: E. I. Khukhro, “Nilpotent length of a finite group admitting a Frobenius group of automorphisms with fixed-point-free kernel”, Algebra Logika, 49:6 (2010), 819–833; Algebra and Logic, 49:6 (2010), 551–560

Citation in format AMSBIB
\Bibitem{Khu10} \by E.~I.~Khukhro \paper Nilpotent length of a~finite group admitting a~Frobenius group of automorphisms with fixed-point-free kernel \jour Algebra Logika \yr 2010 \vol 49 \issue 6 \pages 819--833 \mathnet{http://mi.mathnet.ru/al469} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2829610} \transl \jour Algebra and Logic \yr 2010 \vol 49 \issue 6 \pages 551--560 \crossref{https://doi.org/10.1007/s10469-011-9117-x} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000288430700006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79952243071} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. E. I. Khukhro, “Automorphisms of finite $p$-groups admitting a partition”, Algebra and Logic, 51:3 (2012), 264–277
2. Khukhro E.I., “Fitting Height of a Finite Group with a Frobenius Group of Automorphisms”, J. Algebra, 366 (2012), 1–11
3. N. Yu. Makarenko, E. I. Khukhro, “Lie algebras admitting a metacyclic frobenius group of automorphisms”, Siberian Math. J., 54:1 (2013), 99–113
4. E. I. Khukhro, “Rank and order of a finite group admitting a Frobenius group of automorphisms”, Algebra and Logic, 52:1 (2013), 72–78
5. Guloglu I.S., Ercan G., “Action of a Frobenius-Like Group”, J. Algebra, 402 (2014), 533–543
6. G. Ercan, İ. Güloğlu, E. I. Khukhro, “Rank and order of a finite group admitting a Frobenius-like group of automorphisms”, Algebra and Logic, 53:3 (2014), 258–265
7. Ercan G., Guloglu I.S., “Action of a Frobenius-Like Group With Fixed-Point Free Kernel”, J. Group Theory, 17:5 (2014), 863–873
8. Ercan G., Guloglu I.S., Khukhro E., “Frobenius-Like Groups as Groups of Automorphisms”, Turk. J. Math., 38:6 (2014), 965–976
9. Ercan G., Guloglu I.S., Ogut E., “Nilpotent Length of a Finite Solvable Group With a Frobenius Group of Automorphisms”, Commun. Algebr., 42:11 (2014), 4751–4756
10. de Melo E., “Fitting Height of a Finite Group With a Metabelian Group of Automorphisms”, Commun. Algebr., 43:11 (2015), 4797–4808
11. Ercan G. Guloglu I.S., “On the Influence of Fixed Point Free Nilpotent Automorphism Groups”, Mon.heft. Math., 184:4 (2017), 531–538
12. Ercan G., Guloglu I.S., “Groups of Automorphisms With Tni-Centralizers”, J. Algebra, 498 (2018), 38–46
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